Plebanski's Formulation: Constrained BF Gravity

• Plebanski’s formulation is a reformulation of 4D general relativity as a constrained BF theory using bivector-valued 2-forms and gauge connections.
• It imposes simplicity constraints that ensure the 2-forms originate from tetrads, enabling metric reconstruction via the Urbantke map and alignment with Einstein equations.
• The approach underpins quantum gravity methods, including spin-foam models and pure-connection actions, bridging classical and quantum gravitational frameworks.

Plebanski’s formulation is a reformulation of four-dimensional general relativity as a constrained $BF$ theory. In its non-chiral form the basic variables are a bivector-valued 2-form $B^{IJ}$ and an $SO(1,3)$ or $Spin(4)$ connection; in its chiral form they are a triple of self-dual 2-forms $\Sigma^i$ and an $SO(3,\mathbb{C})$ or $SU(2)$ connection. The defining feature is the simplicity constraint, which forces the 2-form data to arise from a tetrad, after which the metric can be reconstructed by the Urbantke map and the field equations become Einstein or Einstein–Cartan equations in the gravitational sector. The same framework underlies spin-foam models, pure-connection actions, heavenly equations for self-dual metrics, and several matter-coupled and unimodular variants (Tennie et al., 2010, Gonzalez et al., 2012, Bhoja et al., 2024).

1. Constrained $BF$ structure and simplicity constraints

In a standard non-chiral Lorentzian normalization, the Plebanski variables are an antisymmetric 2-form $B^{IJ}$, an $SO(1,3)$ connection one-form $B^{IJ}$0, and a Lagrange multiplier $B^{IJ}$1. The action with cosmological constant is

$B^{IJ}$2

with $B^{IJ}$3. In the chiral $B^{IJ}$4 version one writes instead

$B^{IJ}$5

with symmetric traceless $B^{IJ}$6 (Tennie et al., 2010).

The simplicity constraints are the algebraic conditions that distinguish gravity from unconstrained $B^{IJ}$7 theory. In non-chiral form they may be written as

$B^{IJ}$8

while in spinor language they appear as

$B^{IJ}$9

These equations force the 2-forms to be simple, meaning built from tetrads rather than arbitrary bivector data (Tennie et al., 2010).

Classically, the solutions separate into distinct Plebanski sectors. In the quadratic classification one has

$SO(1,3)$0

together with a degenerate sector characterized by vanishing 4-volume density. The sectors $SO(1,3)$1 and $SO(1,3)$2 differ by the internal Hodge star, whereas the degenerate sector does not admit a nondegenerate metric interpretation (Engle, 2011).

A closely related Euclidean $SO(1,3)$3 form used in spin-foam work is the Holst-$SO(1,3)$4 action

$SO(1,3)$5

with momentum

$SO(1,3)$6

This makes the Immirzi parameter $SO(1,3)$7 explicit at the $SO(1,3)$8-theory level and is the starting point for the discrete $SO(1,3)$9 analysis of the EPRL/FK vertex (Engle, 2011).

2. Metric reconstruction and equivalence to Einstein gravity

Once the simplicity constraints hold, the 2-forms determine a tetrad and hence a metric. In chiral notation, given a tetrad $Spin(4)$0 with $Spin(4)$1, the self-dual 2-forms are

$Spin(4)$2

The metric is recovered from the 2-forms by the Urbantke formula; one representative form is

$Spin(4)$3

Thus the metric is not fundamental but reconstructed from a triple of 2-forms (Gielen et al., 2023, Krasnov et al., 2024).

Varying the action with respect to the connection gives the compatibility equation

$Spin(4)$4

In the gravitational sector this identifies the gauge connection with the self-dual part of the spin connection compatible with the tetrad. In the complex $Spin(4)$5 treatment this relation is

$Spin(4)$6

A central point of the gauge-versus-spacetime connection analysis is that this identification is not automatic from the Plebanski equations alone; it follows once the Urbantke metric and Cartan structure equations are used to relate the internal bundle to spacetime geometry. Under those hypotheses, the internal connection depends only on the Levi-Civita part, independently of torsion and non-metricity that may be present in an arbitrary spacetime connection used in the intermediate derivation (Gonzalez et al., 2012).

Variation with respect to the 2-forms gives the curvature equation

$Spin(4)$7

or, in Einstein form,

$Spin(4)$8

Here the symmetric tracefree field $Spin(4)$9 becomes the self-dual Weyl curvature on shell. In the $\Sigma^i$0-structure formulation one may write

$\Sigma^i$1

so the Einstein condition is equivalent to the statement that the anti-self-dual part of $\Sigma^i$2 vanishes and the scalar part is fixed (Bhoja et al., 2024).

The canonical $\Sigma^i$3 decomposition recovers the complex Ashtekar phase space. With

$\Sigma^i$4

the fundamental Poisson bracket is

$\Sigma^i$5

and the first-class constraints are the Gauss, diffeomorphism, and Hamiltonian constraints. This is the standard canonical bridge from the covariant Plebanski action to Ashtekar’s variables (Gielen et al., 2024).

3. Plebanski sectors, discretization, and spin-foam quantization

In spin-foam applications one discretizes the continuum theory on a triangulation, often focusing on a single oriented 4-simplex. The continuum 2-form is integrated over triangles to give bivectors $\Sigma^i$6, or $\Sigma^i$7 in Holst-$\Sigma^i$8 variables, and two discrete conditions are fundamental: orientation,

$\Sigma^i$9

and closure,

$SO(3,\mathbb{C})$0

For such data there exists a unique constant continuum 2-form $SO(3,\mathbb{C})$1 on an embedded simplex $SO(3,\mathbb{C})$2 reproducing the discrete bivectors, which makes it meaningful to speak of the Plebanski sector of discrete data (Engle, 2011).

A crucial result for the EPRL/FK models is that the linear simplicity constraints do not isolate a single gravitational sector. In a time gauge they imply

$SO(3,\mathbb{C})$3

so the discrete bivectors can be parameterized by areas and 3D normals. However, the stationary-point analysis shows that linear simplicity admits precisely three sectors: $SO(3,\mathbb{C})$4, $SO(3,\mathbb{C})$5, and the degenerate sector. For nondegenerate critical data one has

$SO(3,\mathbb{C})$6

so $SO(3,\mathbb{C})$7 gives sector $SO(3,\mathbb{C})$8 and $SO(3,\mathbb{C})$9 gives sector $SU(2)$0; if the two chiral transport sets are equivalent, the data are degenerate. The resulting asymptotics attribute the two Regge terms $SU(2)$1 to $SU(2)$2, while the extra $SU(2)$3-dependent terms $SU(2)$4 and vector-geometry contributions come from the degenerate sector. The paper explicitly argues that these extra terms arise from sector mixing rather than a sum over manifold orientations (Engle, 2011).

The degenerate sector is independently important. In the $SU(2)$5 theory with the usual trace condition on the Lagrange multiplier removed, the degenerate simplicity constraints are exactly solvable and split into “deg-gravitational” and “deg-topological” subsectors,

$SU(2)$6

with $SU(2)$7. The deg-gravitational subsector reduces to covariantly embedded $SU(2)$8 $SU(2)$9 theory, and its spin-foam quantization is the $BF$0 Crane–Yetter state sum (Alexandrov, 2012).

This solvable sector also clarifies a quantization issue. Restricting $BF$1 representations and intertwiners, the usual strategy in EPRL/FK, is not sufficient to recover the correct vertex amplitude. The proposed remedy is to impose the secondary second-class constraints in the vertex measure on an equal footing with the primary simplicity constraints. In the degenerate model this produces the $BF$2 $BF$3 vertex and the full $BF$4 Crane–Yetter/Ooguri state sum, in agreement with canonical path-integral quantization (Alexandrov, 2012).

4. Pure-connection, $BF$5potential, instanton, and unimodular variants

A major development of the Plebanski framework is the elimination of auxiliary fields to obtain pure-connection or $BF$6potential descriptions. For $BF$7, one pure-connection action for self-dual gravity is

$BF$8

and the self-dual gravity truncation takes the form

$BF$9

Variation with respect to $B^{IJ}$0 imposes $B^{IJ}$1, so the metric can again be reconstructed by Urbantke, now directly from curvature rather than from an independent $B^{IJ}$2-field (Krasnov et al., 2024).

A related line of work shows that there is a family of field redefinitions acting on $B^{IJ}$3-type Lagrangians of gravity. In the chiral case this is a one-parameter family; in the non-chiral case it is a two-parameter family. Lagrangians related by these transformations are classically equivalent, differing by topological terms and reparameterizations of auxiliary matrices. In particular, the chiral transformation gives an alternative derivation of the $B^{IJ}$4potential formulation of general relativity, while the non-chiral analysis yields a new $B^{IJ}$5potential form of GR and clarifies the status of non-chiral pure-connection actions (Krasnov, 2017).

Another Plebanski-like action modifies the way the symmetric auxiliary matrix enters: $B^{IJ}$6 Its advantages are that the symmetric matrix can be integrated out, leading to a pure $B^{IJ}$7-type action for GR, and that the special choice $B^{IJ}$8, $B^{IJ}$9 yields conformally anti-self-dual gravity. The canonical analysis shows that the resulting phase space matches the Ashtekar formalism up to a canonical transformation induced by a topological term (Celada et al., 2016).

The instanton representation of Plebanski gravity is another reformulation adapted to the self-dual sector. Its basic fields are a gauge connection and a nondegenerate $SO(1,3)$0 matrix $SO(1,3)$1, with $SO(1,3)$2. In the nondegenerate sector, together with the symmetry condition on $SO(1,3)$3, the algebraic constraint $SO(1,3)$4, and the Gauss law, this is equivalent to Plebanski’s equations. The formulation makes Hodge self-duality of $SO(1,3)$5 explicit and is tailored to gravitational instantons (III, 2012).

Unimodular Plebański gravity replaces the fixed cosmological constant parameter by an integration constant. In the fixed-volume version the action is

$SO(1,3)$6

while the parametrized Henneaux–Teitelboim version uses an exact 4-form $SO(1,3)$7 instead of $SO(1,3)$8. In both cases $SO(1,3)$9 is not fixed a priori: in the fixed-volume theory it becomes constant on shell by the Bianchi identity, and in the parametrized theory $B^{IJ}$00 is a field equation. These theories also admit pure-connection reductions, such as

$B^{IJ}$01

with $B^{IJ}$02 (Gielen et al., 2023).

The canonical analysis of the unimodular variants retains the Ashtekar pair $B^{IJ}$03 but changes the status of the Hamiltonian. In the preferred-volume theory the Hamiltonian density is constrained to be spatially constant rather than to vanish, while in the parametrized theory a new canonical pair $B^{IJ}$04 appears, with $B^{IJ}$05 constrained to be spatially and temporally constant. This yields one additional global degree of freedom interpreted as the cosmological constant, together with a natural “volume time” built from the exact 4-form $B^{IJ}$06 (Gielen et al., 2024).

5. Matter couplings, source terms, and torsion

Matter coupling in the Plebanski formalism is less direct than in metric GR because the basic gravitational variables are 2-forms and connections rather than a metric and its Levi-Civita connection. For tensor matter, a general prescription is to take an ordinary matter action $B^{IJ}$07 and replace the metric by the Urbantke metrics built from the self-dual and anti-self-dual 2-forms: $B^{IJ}$08 For Dirac spinors one reconstructs a tetrad from the Urbantke metric and inserts it into the Einstein–Cartan Dirac action. In the gravitational sector these prescriptions reproduce exactly the Einstein–Cartan equations with matter; for bosonic matter they give the usual torsion-free coupling, while for fermions the torsion equation is algebraic and sourced by the spin current (Tennie et al., 2010).

The same paper verifies concrete scalar, Yang–Mills, and Dirac couplings. For Yang–Mills, an auxiliary-field action based on $B^{IJ}$09, $B^{IJ}$10, and symmetric spinors $B^{IJ}$11, $B^{IJ}$12 reproduces the standard Yang–Mills stress tensor after eliminating auxiliaries. For spinors, the $B^{IJ}$13-equation becomes the Einstein–Cartan torsion equation and the curvature equations reduce to the Einstein–Cartan relations once the translation formulas between $B^{IJ}$14-based and tetrad-based sources are used (Tennie et al., 2010).

A more direct translation of metric matter sources into chiral Plebanski variables starts from the decomposition

$B^{IJ}$15

and lifts $B^{IJ}$16 into the algebraic curvature space by the Kulkarni–Nomizu product

$B^{IJ}$17

The chiral matter source is then

$B^{IJ}$18

and the matter-coupled field equation becomes

$B^{IJ}$19

Applying the chiral Bianchi identity $B^{IJ}$20 then reproduces the standard conservation law $B^{IJ}$21. For a spherically symmetric Maxwell source, the anti-self-dual part of the equations yields the Reissner–Nordström–de Sitter metric (Hughes et al., 18 Jun 2026).

Torsion occupies a special place in the non-chiral formulation. In the ordinary gravitational sector, if $B^{IJ}$22 or its dual, then the connection equation

$B^{IJ}$23

implies

$B^{IJ}$24

and for nondegenerate tetrads this gives $B^{IJ}$25. However, through the Nieh–Yan identity

$B^{IJ}$26

a torsion-squared action can be related to the $B^{IJ}$27 term up to a boundary contribution. In the self-dual setting this leads to the statement that a torsionful phase is equivalent to ordinary gravity or to a topological $B^{IJ}$28 phase, so torsion need not be treated as an independent propagating variable in these formulations (Bennett et al., 2012).

6. Self-dual sector, heavenly equations, and later developments

The self-dual sector of Plebanski theory admits a remarkable scalar reduction. In the second heavenly formulation, the problem of finding self-dual Einstein metrics is reduced to a nonlinear second-order PDE for a single potential. In one standard coordinate form the equation is

$B^{IJ}$29

A recent pure-connection derivation obtains both the flat-space version and the constant-curvature analogue by using a complex basis of self-dual 2-forms and a covariant light-cone ansatz for the chiral connection, showing that the heavenly equation emerges directly from the self-dual gravity equations written in pure-connection form (Krasnov et al., 2024).

The same pure-connection analysis emphasizes structural parallels between self-dual gravity and self-dual Yang–Mills. In particular, the vector fields

$B^{IJ}$30

satisfy

$B^{IJ}$31

which gives a concrete realization of the self-dual Yang–Mills kinematic algebra as the Lie algebra of $B^{IJ}$32 vector fields on $B^{IJ}$33 endowed with a complex structure. This provides a direct bridge between Plebanski’s self-dual gravity, pure-connection methods, and color–kinematics structures (Krasnov et al., 2024).

The second heavenly equation also supports a perturbiner expansion for self-dual spacetimes. In the marked-tree formulation, a self-dual background with arbitrarily many positive-helicity gravitons is constructed as a sum over tree graphs, while negative-helicity gravitons are added as linear anti-self-dual perturbations. Evaluating the self-dual Plebanski action supplemented by a boundary term on this background produces the graviton MHV tree amplitude. A distinctive feature of that derivation is that the amplitude comes entirely from the boundary term and reproduces the NSVW tree formula without using BCFW recursion or twistor theory (Miller, 2024).

More recent work extends the chiral differential-form viewpoint beyond formal equivalence statements. The $B^{IJ}$34-bundle interpretation treats the basic 2-forms as soldering forms on an associated bundle, develops linearized and nonlinear gauge fixings for the Einstein equations in chiral variables, and uses the self-dual/anti-self-dual decomposition to analyze type $B^{IJ}$35 spacetimes and evolution schemes for numerical relativity. A plausible implication is that Plebanski’s formulation is not only a reformulation of classical GR but also a flexible organizational framework for canonical, covariant, integrable, and numerical approaches (Shaw, 22 Apr 2026).

Taken together, these developments show that Plebanski’s formulation is simultaneously a constrained $B^{IJ}$36 theory, a route to Ashtekar variables, a foundation for spin-foam amplitudes, a source of pure-connection and $B^{IJ}$37potential actions, and a natural language for self-dual geometry. Its central operation remains unchanged across these contexts: replacing the metric by 2-form data, and recovering Einstein geometry through simplicity, compatibility, and curvature equations.